Is a limit a variable or a constant? Is a limit an exact value or approximate value?

My current understanding of the limit is that it is a constant based on applications related to physics such as terminal velocity. Also, I believe that since a limit can possibly be infinity, that limits are not exact values. Please tell me how I can understand this better. Thanks.

3 Answers
3

There are many things that limits can be (many settings in which the notion of "limit" makes sense), which makes your question very difficult to answer succintly.

Since you label this [calculus], I'm going to assume that you are talking about limits of real functions of real variable.

That is: we have a function $f\colon D\to\mathbb{R}$ (where $D\subseteq\mathbb{R}$ is the domain of $f$), and we are considering something like $\lim\limits_{x\to a}f(x)$ where $a$ is a real number. (I'll discuss weakening these conditions after we're done with this basic case).

In this case, the limit of a real-valued function at a real point $a$ can either exist or not exist. If it exists, it will be a number (a value, a constant, a specific real number, an exact, specific, single real number).

The precise definition is the following: first, we can only talk about "the limit of $f(x)$ as $x$ approaches $a$" if there is an interval $(c,d)$, with $c\lt a \lt d$, such that every $x$ in $(c,d)$, except perhaps for $a$, is in the domain of $f$. In that case, we say that the limit of $f(x)$ as $x$ approaches $a$ is equal to the number $L$ if and only if the values of $f(x)$ are arbitrarily close to $L$ for all $x$ in $(c,d)$ that is sufficiently close, but not equal to, $a$.

That's a little fuzzy; what it formally means is: if you specify how close you want the values of $f(x)$ to be to $L$ (which is done by specifying a distance $\epsilon\gt 0$ that we are not allowed to exceed), we can specify a distance $\delta\gt 0$ so that if $x$ is any number in $(c,d)$, other than $a$, which is within a distance of $\delta$ from $a$, then $f(x)$ will be within a distance of $\epsilon$ to $L$.

It may be possible that no such $L$ exists. But if it does exist, then we write
$$\lim_{x\to a}f(x) = L.$$
Now, $L$ is a specific, exact, single number (one can show that it is impossible for two different numbers $L_1$ and $L_2$ to both be limits of $f(x)$ as $x$ approaches $a$).

Now, the values that $f(x)$ takes when $x$ is close to, but not equal to, $a$, need not be exactly equal to $L$; the value at $a$ need not have anything to do with $L$. But the limit is exactly $L$. It doesn't change, it doesn't move, it's not approximate. We are talking about what the values of $f(x)$ approach. (Just like how far you are from your bed doesn't change the fact that your bed is in a precise, specific location in your room).

The limit is an exact value, period. Because the limit describes what the values of the function are approaching, and they can only approach one thing. (You cannot simultaneously be getting arbitrarily close to two different places, because at some point, in order to approach place A more, you'll start getting farther away from B).

Now, it's possible that such a number $L$ does not exist. One possible reason for this may be that as we get closer and closer to $a$, the values of $f(x)$ get larger, and larger, and larger, without bound. (For exmaple, is $f(x) = \frac{1}{x^2}$ and $x$ gets closer to $0$). In that situation, we want to specify why the limit, as $x$ approaches $0$, does not exist. When we write
$$\lim_{x\to 0}\frac{1}{x^2}=\infty$$
we are not saying that the limit actualy takes the value "infinity", we are really saying two things:

The limit does not exist; and

Why the limit does not exist (namely, the values of the function grow without bound as $x$ gets closer to $0$).

(Similarly when we say the limit is $-\infty$, we are saying the limit does not exist, and we are saying why it does not exist: because the values are getting large and negative without bound).

Then there's the concept of "limit as $x\to\infty$". Here, we are trying to discuss what happens to the values of the function as $x$ gets larger, and larger, and larger. Again, the limit may exist or not exist. The precise definition is very similar to the definition given above for a regular limit, but instead of "provided $x$ is sufficiently close, but not equal to, $a$", we say "provided that $x$ is sufficiently large", and we require that $f$ be defined on some interval of the form $(b,\infty)$. That is:

The limit of $f(x)$ as $x\to\infty$ is the number $L$ if and only if the values of $f(x)$ get arbitrarily close to $L$ for any sufficiently large $x$; that is, if you specify a distance $\epsilon$ from $L$, then we can find a value $M\gt 0$ such that every $x\gt M$ will have the property that $f(x)$ is within $\epsilon$ of $L$.

Again, this $L$ is a real number, it is unique, it is exact, and it is not an approximations. The value of $f(x)$ may or may not be equal to $L$, but they will definitely fall within $\epsilon$ of $L$ as $x$ gets larger.

Again, we sometimes say things like "$\lim\limits_{x\to\infty}f(x) =\infty$", but what we mean is that the limit does not exist, but we are saying why it does not exist.

In summary: for real valued functions, when a limit exists, it is a unique real number: precise and exact.

Now, there are other situations in which we can talk about limits. We can definitely talk about limits any time we can talk about distances between the images. This occurs with functions that take vector values, and in many other situations. We can even talk about limits in situations where there is no sense of "distance", but there is a sense of "nearness" (this is covered in the mathematical study of topology.

Also, we can sometimes try to compute several limits at the same time. This is what happens when you look at the derivative function, for example: the derivative of $f(x)$ at $x$ is
$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h},$$
and it seems like the answers are not numbers. In fact, they are numbers, it's just that we are doing all the limits at the same time, and giving a formula for their answers.

A limit really isn't a variable or a constant. It is normally an exact value, except in cases where the limit has to be approximated. (I can't quite think of any cases like that, but they probably exist.)

But the real question here lies in what you mean by "exact value". Infinity, in the context of limits, is considered an exact value in that it is the 'exact' answer to, say: $\lim_{n \to \infty} n$ or $\lim_{n \to \infty} \frac{1}{n^2}$. Limits have exact values (I would prefer to say "answers") if they exist.

However, sometimes limits are simply undefined. For example, $lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{k}$. A more obvious example would be $\lim_{n \to 1}\frac{1}{x-1}$. In cases like that, it becomes helpful to define left and right limits.

Instead of considering $\lim_{n \to 1}\frac{1}{x-1}$, it is more constructive to consider its respective left and right limits: $\lim_{n \to 1^{-}}\frac{1}{x-1}$ and $\lim_{n \to 1^{+}}\frac{1}{x-1}$. These are not undefined; they have answers: $-\infty$ and $\infty$, respectively.

I'm not particularly acquainted with the applications of limits to Physics, so I can't quite help you there. But I hope this provides you a very rough and simple overview of what a limit is.

As a side note, I think there are cases where limits simplify to variables, but I'll leave that to the more capable users here.

If you have a problem where something approaches a finite number, I think the best answer to your question is "The limit is a constant."

If you have a problem where something approaches infinity, I think the best answer to your question is "In that problem, the limit doesn't exist."

Let's consider, for example, the statement
$$
\lim_{x\to3} \frac{x^2-9}{x-3}=6.
$$
According to the definitions traditionally used, this is a true statement. (Because one can remove a common factor of $x-3$.)

In my opinion, if you want to talk about the above limit in an informal or intuitive way, you should probably say that the limit is exactly 6.

If discussing this limit problem in an informal or intuitive way, one might focus on the idea that something's approaching 6, but I think it's important to remember that the exact value of the limit is the unique number 6.

If you're talking about limits, you're already in some sense talking about approach. You don't need to say "the limit approaches 6"; it's fine to say "the limit IS 6".

I would disagree that the reason the statement you have is true is "one can remove a common factor of $x-3$". The reason the statement is true is that for every $\epsilon\gt 0$ there exists $\delta\gt 0$ such that for all $x$, if $0\lt |x-3|\lt \delta$, then $|\frac{x^2-9}{x-3} - 6|\lt\epsilon$. And this can be justified by invoking several theorems, e.g., that two functions with the exact same value verywhere except at $3$ have the same limit as $x\to 3$ (which is what allows you to "remove a common factor" above), and some others...
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Arturo MagidinJan 15 '12 at 1:23